A rational function is defined as this:
$$
f(X) = \frac{a_0 + a_1X + a_2x^2 + \cdots a_n X^n }
{b_0 + b_1X + b_2 X^2 + \cdots + b_mX^M}
$$
Which allowing complex numbers can be factored to this:
$$
f(X) = \frac{a (X - \alpha_1)^{e_1} (X - \alpha_2)^{e_2} \cdots (X - \alpha_r)^{e_r}}
{b (X - \beta_1)^{d_1} (X - \beta_2)^{d_2} \cdots (X - \beta_s)^{d_s}}
$$
And has a "divisor" defined as this:
$$
\operatorname{div}(f(X)) = e_1[\alpha_1] + e_2[\alpha_2] + \cdots + e_r[\alpha_r] - d_1[\beta_1] - d_2[\beta_2] - \cdots - d_r[\beta_r]
$$
The first two steps make perfect sense. The last step doesn't make sense at all. How does one get from the polynomial fraction to the divisor? Can someone explain? Also, in the divisor equation, why does the equation go from $d_1[\beta_1]$ to $d_r[\beta_r]$ rather than $d_s[\beta_s]$
To understand how one gets from the polynomial fraction to the divisor, let's break it down step by step:
1. Start with the rational function in its factored form:
$$
f(X) = \frac{a (X - \alpha_1)^{e_1} (X - \alpha_2)^{e_2} \cdots (X - \alpha_r)^{e_r}}
{b (X - \beta_1)^{d_1} (X - \beta_2)^{d_2} \cdots (X - \beta_s)^{d_s}}
$$
2. Notice that in the factored form, the numerator represents the factors that make the function zero (zeros), and the denominator represents the factors that make the function undefined (poles or vertical asymptotes).
3. Now, let's focus on the divisor equation:
$$
\operatorname{div}(f(X)) = e_1[\alpha_1] + e_2[\alpha_2] + \cdots + e_r[\alpha_r] - d_1[\beta_1] - d_2[\beta_2] - \cdots - d_r[\beta_r]
$$
4. In the divisor equation, the terms with $[\alpha]$ represent the multiplicities of the zeros of the function (i.e., $\alpha_i$) in the numerator, and the terms with $[\beta]$ represent the multiplicities of the poles or vertical asymptotes of the function (i.e., $\beta_i$) in the denominator.
5. The multiplicities $e_i$ and $d_i$ indicate how many times each zero or pole occurs in the factorization. For example, if $(X - \alpha_i)^{e_i}$ appears in the numerator, it means that $\alpha_i$ is a zero with multiplicity $e_i$. Likewise, if $(X - \beta_i)^{d_i}$ appears in the denominator, it means that $\beta_i$ is a pole with multiplicity $d_i$.
6. The equation starts with $e_1[\alpha_1]$ and goes up to $e_r[\alpha_r]$ because there might be multiple zeros in the numerator, each with its own multiplicity. Similarly, the equation goes from $d_1[\beta_1]$ to $d_r[\beta_r]$ because there might be multiple poles in the denominator, each with its own multiplicity.
In summary, the divisor equation provides a concise representation of the zeros and poles of a rational function, taking into account their multiplicities. It allows you to understand the behavior of the function and analyze its graph and asymptotic behavior.
To understand how to go from the rational function to the divisor, let's break down each component of the divisor equation:
1. The terms with exponents $e_1, e_2, \ldots, e_r$ represent the powers to which the factors $(X - \alpha_1), (X - \alpha_2), \ldots, (X - \alpha_r)$ are raised in the numerator of the rational function.
2. The terms with exponents $d_1, d_2, \ldots, d_s$ represent the powers to which the factors $(X - \beta_1), (X - \beta_2), \ldots, (X - \beta_s)$ are raised in the denominator of the rational function.
3. The notation $[\alpha]$ represents the point at which the polynomial factor $(X - \alpha)$ is equal to zero. In other words, $[\alpha]$ is the root or zero of the polynomial factor.
Now, as for the discrepancy in the divisor equation, where the equation goes from $d_1[\beta_1]$ to $d_r[\beta_r]$ rather than $d_s[\beta_s]$, it appears to be a typographical error in the statement you provided.
The correct representation should be:
$$\operatorname{div}(f(X)) = e_1[\alpha_1] + e_2[\alpha_2] + \cdots + e_r[\alpha_r] - d_1[\beta_1] - d_2[\beta_2] - \cdots - d_s[\beta_s]$$
where the terms with exponents $d_1, d_2, \ldots, d_s$ represent the powers of the factors in the denominator of the rational function.
In summary, to obtain the divisor from the rational function, you factor the numerator and denominator into their prime factors and compare the exponents of each factor. The divisor equation then represents the sum of these exponents multiplied by the corresponding roots of the factors.